No, throwing your mattress or furniture away will not solve a bed bug problem. If you have bed bugs in your mattress or couch, they are also living in other places in your house. Any new furniture that you bring into your home will also get bed bugs.
Ask your pest control operator if there is anything you need to throw away. Even if furniture such as a couch has bed bugs, the pest control operator may be able to treat it. If you do need to throw something away, spray paint or write “Bed Bugs” on the item in large letters so that no one picks it up and brings it into their home. Do not get new furniture until the pest control operator has finished treating your home and you are sure that bed bugs are gone.